Chapter 7 Functions

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Chapter 7 Functions

• Dr. Curry Guinn

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Outline of Today

• Section 7.1 Functions Defined on General Sets
• Section 7.2 One-to-One and Onto
• Section 7.3 The Pigeonhole Principle
• Section 7.4 Composition of functions
• Section 7.5 Cardinality

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What is a function?

• A function f f X ? Y
• Maps a set X to a set Y
• is a relation between
• the elements of X (called the inputs) and
• the elements of Y (called the outputs)
• with the property that each input is related to
  one and only one output.
• X is the domain.
• Y is the co-domain.
• The set of all values f(x) is called the range.

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How do we represent functions

• Arrow diagrams
• f X ? Y

• What is the domain?
• a,b,c,d
• What is the co-domain?
• x, y, z, p, q, r, s
• What is the range?
• x, y, p
• What is the inverse image of y?
• a, c

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How do we represent functions?

• As Ordered Pairs
• f (a,y), (b,p), (c,y), (d,x)

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How do we represent functions?

• As machines

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How do we represent functions?

• By Formula
• f(x) 2x2 3

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Equality of functions

• Two functions, f and g, are equal if
• Both map from set X to set Y
• And
• f(x) g(x) for all x ? X.
• If f(x) SQRT(x2) and g(x) x, is f g?
• Identity function, i is such that
• i(x) x for all x ? X

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The Logarithmic Function

• Logb x y ? by x
• Log2 8
• Log10 1000
• Log3 3n
• Log5 1/25
• Loga 1 for a gt 0

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Well defined function

• Remember a function must map an input to a
  single, unique value
• F R ? R such that
• f(x) SQRT(-x2) for all real numbers X.
• Why is this not well defined?

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7.2 One-to-one and onto

• A function is f X ? Y is one-to-one when
• If f(x1) f(x2), then x1 must be equal to x2.
• To show a function is one-to-one, assume f(a)
  f(b) for arbitrary a and b. Show a b.

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One-to-one and finite sets

• See board

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One-to-one example, infinite sets

• f(n) 2n 1
• f(n) n2

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Onto functions

• A function f X ? Y is onto if for every y in
  the co-domain, that is, every y ? Y, there exist
  some x ? X, such that f(x) y.
• To prove something is onto, pick an arbitrary
  element in Y and find an x in X that maps to the
  y.

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Onto functions and finite sets

• See board

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Onto examples, infinite sets

• Show the following are or are not onto
• f Z ? Z by f(n) 2n 1.
• f Z ? Z by f(n) n 5.

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One-to-one correspondences

• If a function f X ? Y is both one-to-one and
  onto, there is a one-to-one correspondence (or
  bijection) from the set X to set Y.
• Show f Z ? Z by f(n) n 5 has a one-to-one
  correspondence.

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Inverse functions

• If a function f X ? Y is has as one-to-one
  correspondence, the there is an inverse function
  f-1 Y ? X such that if f(x) y, then f-1(y)
  x.
• f(x) n 5
• Whats the inverse?

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Exponential and Logarithmic functions

• Expb(x) bx for any x ? R and b gt 0.
• Logb(x) y, for any x ? R if x by
• Show logb(x/y) logb(x) logb(y)
• Hint Let u logb(x) and v logb(y)

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7.3 The Pigeonhole Principle

• Suppose X and Y are finite sets and N(X) gt N(Y).
  Then a function f X ? Y cannot be one-to-one.
• Proof by contradiction.

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Using the Pigeonhole Principle

• Prove there must be at least 2 people in New York
  city with the same number of hairs on their head.
• How many integers must you pick in order for them
  to have at least one pair with the same remainder
  when divided by 3.

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The Generalized Pigeon Hole Principle

• For any function f from a finite set X to a
  finite set Y and for any positive integer k, if
  N(X) gt kN(Y), then there is some y ? Y such that
  the inverse image of y has at least k1 distinct
  elements of X.
• If you have 85 people, and there are 26 possible
  initials of their last name, at least one initial
  must be used at least ___ times.

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Pigeonhole

• In a group of 1,500 people, must at least five
  people have the same birthday?

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7.4 Composition of Functions
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Composition of functions

• The composition of two functions occurs when the
  output of one function is the input to another.
• Let f X ? Y and g Y ? Z where the range of f
  is a subset of the domain of g. Define a new
  function g ? f(x) g(f(x)).

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Composition Examples

• F(n) n 1 and g(n) n2
• What is f ? g?
• What is g ? f?

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Composition of one-to-one functions

• If both f and g are one-to-one, is f ? g
  one-to-one?
• Proof See board
• If both f and g are onto, is f ? g onto?

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7.5 Cardinality

• The cardinality of a set is how many members it
  has.
• Let X and Y be sets. X has the same cardinality
  as Y iff there exists a one-to-one correspondence
  from X to Y.
• X has the same cardinality as X (Reflexive)
• If X has the same cardinality as Y, Y has the
  same cardinality as X (Symmetric)
• If X has the same cardinality as Y, and Y has the
  same cardinality as Z, then x has the same
  cardinality as Z (Transitive).

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Countable Sets

• A set X is countably infinite iff has the same
  cardinality as the set of positive integers.
• Is the set of all integers countable?
• F(n)
• 0 if n 1
• -n/2 if n is even
• (n-1)/2 if n is odd
• Is this one-to-one? Onto?

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Rational numbers are countable
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The set of real numbers between 0 and 1 is
uncountable.

• Uses Cantors Diagnolization Argument
• Proof by contradiction
• See board!!
• Any set with an uncountable subset is
  uncountable.

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Some interesting results

• The set of all computer programs in a given
  computer language is countable.
• How?
• Each program is a finite set of strings.
• Convert to binary.
• Now each program is a unique number in the set of
  integers.
• The set of all programs is a subset of the set of
  all integers. Therefore countable.